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Thin monodromy in Sp(4)

  • Christopher Brav (a1) and Hugh Thomas (a2)

Abstract

We show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$ . In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$ . As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.

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References

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[BH89]Beukers, F. and Heckman, G., Monodromy for the hypergeometric function nF n−1, Invent. Math. 95 (1989), 325354; MR 974906 (90f:11034).
[CK08]Canonaco, A. and Karp, R. L., Derived autoequivalences and a weighted Beilinson resolution, J. Geom. Phys. 58 (2008), 743760.
[CYY08]Chen, Y.-H., Yang, Y. and Yui, N., Monodromy of Picard–Fuchs differential equations for Calabi–Yau threefolds, J. Reine Angew. Math. 616 (2008), 167203; with an appendix by Cord Erdenberger; MR 2369490 (2009m:32046).
[DM86]Deligne, P. and Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 589; MR 849651 (88a:22023a).
[DM06]Doran, C. F. and Morgan, J. W., Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi–Yau threefolds, in Mirror symmetry. V, AMS/IP Studies in Advanced Mathematics, vol. 38 (American Mathematical Society, Providence, RI, 2006), 517537; MR 2282973 (2008e:14010).
[ES06]van Enckevort, C. and van Straten, D., Monodromy calculations of fourth order equations of Calabi–Yau type, in Mirror symmetry. V, AMS/IP Studies in Advanced Mathematics, vol. 38 (American Mathematical Society, Providence, RI, 2006), 539559; MR 2282974 (2007m:14057).
[FMS13]Fuchs, E., Meiri, C. and Sarnak, P., Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions, Preprint (2013), arXiv:1305.0729.
[GS75]Griffiths, P. and Schmid, W., Recent developments in Hodge theory: a discussion of techniques and results, in Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) (Oxford University Press, Bombay, 1975), 31127; MR 0419850 (54 #7868).
[Kuz04]Kuznetsov, A. G., Derived category of a cubic threefold and the variety $V_{14}$, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 183–207;MR 2101293 (2005i:14049).
[LW85]Lee, R. and Weintraub, S. H., Cohomology of Sp4(Z)and related groups and spaces, Topology 24 (1985), 391410; MR 816521 (87b:11044).
[LS77]Lyndon, R. C. and Schupp, P. E.,Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89 (Springer, Berlin, 1977); MR 0577064 (58 #28182).
[Nor86]Nori, M. V., A nonarithmetic monodromy group, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 7172; MR 832040 (87g:14008).
[Sar12]Sarnak, P., Notes on thin matrix groups, Preprint (2012), arXiv:1212.3525.
[SV12]Singh, S. and Venkataramana, T. N., Arithmeticity of certain symplectic hypergeometric groups, Preprint (2012), arXiv:1208.6460.
[Ste+12]Stein, W. A. et al. , Sage mathematics software (Version 5.3), The Sage Development Team, 2012, http://www.sagemath.org.
[Swa69]Swan, R. G., Groups of cohomological dimension one, J. Algebra 12 (1969), 585610; MR 0240177 (39 #1531).
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Thin monodromy in Sp(4)

  • Christopher Brav (a1) and Hugh Thomas (a2)

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